A right-angle grinder with a diamond blade would be the tool-of-choice for a professional installer but this isn't a tool a homeowner doing a one-time tile installation is likely to have available to them.

The floor vent would be easy enough to cut if more than one tile was involved because you can measure and mark your tiles and cut from the edges, your wet saw would do the trick.

IF, the required hole falls inside a tile and you can't cut from the edges you must "plunge-cut" with your saw. You would accurately mark your cut locations then (depending on the type of saw you are using) "carefully plunge" the blade into the tile or "carefully plunge" the tile into the saw blade. Either way will work.

Once you make your top-cuts to the corner marks for accuracy, you can then turn the tile over to complete the cuts into the corners. Cuts on the back of the tile can be slightly "over-cut" if need be to complete a cut.

The more you can cut with the saw, the less likely you are to crack the tile when finally removing the waste.

If this isn't clear to you say so and I'll try to clear it up for you.

If you require curved cuts, then that would be a floor vent cut requirement I have never seen, but it could be done.